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2020 Convergence analysis of Tikhonov regularization for non-linear statistical inverse problems
Abhishake Rastogi, Gilles Blanchard, Peter Mathé
Electron. J. Statist. 14(2): 2798-2841 (2020). DOI: 10.1214/20-EJS1735


We study a non-linear statistical inverse problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization) approach to estimate the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the concept of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.


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Abhishake Rastogi. Gilles Blanchard. Peter Mathé. "Convergence analysis of Tikhonov regularization for non-linear statistical inverse problems." Electron. J. Statist. 14 (2) 2798 - 2841, 2020.


Received: 1 November 2019; Published: 2020
First available in Project Euclid: 7 August 2020

zbMATH: 07235727
MathSciNet: MR4132644
Digital Object Identifier: 10.1214/20-EJS1735

Primary: 65J20
Secondary: 62G08 , 62G20 , 65J15 , 65J22

Keywords: general source condition , Minimax convergence rates , ‎reproducing kernel Hilbert ‎space , Statistical inverse problem , Tikhonov regularization


Vol.14 • No. 2 • 2020
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